Problem: Let $a,$ $b,$ and $c$ be the roots of $x^3 - 7x^2 + 5x + 2 = 0.$  Find
\[\frac{a}{bc + 1} + \frac{b}{ac + 1} + \frac{c}{ab + 1}.\]
Solution: By Vieta's formulas, $a + b + c = 7,$ $ab + ac + bc = 5,$ and $abc = -2.$

We can say
\[\frac{a}{bc + 1} + \frac{b}{ac + 1} + \frac{c}{ab + 1} = \frac{a^2}{abc + a} + \frac{b^2}{abc + b} + \frac{c^2}{abc + c}.\]Since $abc = -2,$ this becomes
\[\frac{a^2}{a - 2} + \frac{b^2}{b - 2} + \frac{c^2}{c - 2}.\]By Long Division, $\frac{x^2}{x - 2} = x + 2 + \frac{4}{x - 2},$ so
\begin{align*}
\frac{a^2}{a - 2} + \frac{b^2}{b - 2} + \frac{c^2}{c - 2} &= a + 2 + \frac{4}{a - 2} + b + 2 + \frac{4}{b - 2} + c + 2 + \frac{4}{c - 2} \\
&= a + b + c + 6 + 4 \left( \frac{1}{a - 2} + \frac{1}{b - 2} + \frac{1}{c - 2} \right) \\
&= 7 + 6 + 4 \cdot \frac{(b - 2)(c - 2) + (a - 2)(c - 2) + (a - 2)(b - 2)}{(a - 2)(b - 2)(c - 2)} \\
&= 13 + 4 \cdot \frac{(ab + ac + bc) - 4(a + b + c) + 12}{abc - 2(ab + ac + bc) + 4(a + b + c) - 8} \\
&= 13 + 4 \cdot \frac{5 - 4 \cdot 7 + 12}{-2 - 2 \cdot 5 + 4 \cdot 7 - 8} \\
&= \boxed{\frac{15}{2}}.
\end{align*}